MODELLING DIFFERENTIAL PAIRS FOR LOW-DISTORTION AMPLIFIER DESIGN
E. K. de Lange, 0.De Fe0
A. van Staveren
Swiss Federal Institute of
Technology Lausanne
CH- 1015 Ecublens, Switzerland
Faculty of ITS
Delft University of Technology
Mekelweg 4,2628CD Delft, Netherlands
ABSTRACT
The differential pair is one of the basic amplifying stages in amplifier design. Aiming at the the design of low-distortion amplifiers, a
simple model for a differential pair is derived using the Volterra series. The parameters in this model correspond to the well-known
small-signal parameters, furthermore the effect of the non-ideal
tail-current source is made explicit in the model. The model gives
insight in the way distortion is generated in a differential pair and
it therefore allows the establishment of some simple design rules.
Although being derived from a very simple model, the expressions
give a good accordance with simulations.
1. INTRODUCTION
The accuracy of high-performance negative-feedback amplifiers
depends on three factors: noise, bandwidth and distortion [I]. Much
is known about the effects of noise and bandwidth and how to optimize them. However the treatment of harmonic distortion in the
same context of structured electronic design has not had as much
attention. The principal reason for this inequality is probably the
fact that whereas tools have been developed to analyze the distoTtion for a given configuration [2], [3], less is known about the
possibility to develop synthesis tools and N k S for low-distortion
design.
A generic block scheme for a negative-feedback amplifier is
shown in Fig. 1. Ideally, when an infinitely large amplification can
be given by the active part A (the nullor approximation), there is
no distortion [I]. In a first approach, the design is concentrated
on realizing an accurate transfer through the passive feedback 8.
Nowadays, however, with low-power constraints and RF applications making it more difficult to guarantee high amplification, designers get interested in getting insight in the way distortion arises
in active amplifier circuits.
Figure I: A negative-feedback system
The two most basic amplifying stages in amplificr design, because of their high amplification, are the Common Emitter (CE)
stage and it's balanced counterpart the Differential Pair (DP) [I].
R7803-7761-31031$17.0002003 IEEE
1.261
The design of the active parl A can then be considered as the cascading of an appropriate number of C E D P stages (see Fig. I).
This article focuses on the distortion behavior of the differential
pair, the CE-stage having been treated in a previous publication
~41.
This paper aims to develop some design rules regarding distortion in a differential pair. Some authors have published about
the distortion in the differential pair e.g. [ 5 ] , [ 6 ] . In these articles
usually an analysis is given which explains or models a certain effect in the differential pair. The focus ofthis paper is, while taking
into account the effects that were analyzed in these earlier papers,
to come to the simplest possible model such as to provide a maximum of insight and to determine simple rules for the synthesis of
a differential pair with low distortion.
Because one wants a high-performance amplifier to be as linear as possible, only weakly nonlinear phenomena will occur. This
allows the use of Volterra series to obtain closed-form expressions
for the distortion up till the third-order. Higher-order analysis is
possible, but has little use since distortion decays drastically with
the order [3]. An introduction to the use of the Volterra series
for distortion calculations is given in Sec. 2. A (highly simplified) model for the differential pair and its distortion is presented
in Sec. 3. Section 4 presents a discussion of the results and some
design rules. The results are compared to simulations with PStar
(a simulator that uses the harmonic balance method) in Sec. 5. An
excellent agreement is obtained in spite of the drastic simplifications.
2. VOLTERRA SERIES AND DISTORTION
Since the goal in the design of high-performance amplifiers is to
get a close-to-linear transfer, the nonlinearities that will occur will
as a consequence be relatively weak. This fact suggest the Volterra
series as the perfect tool to describe the distortion. Volterra series
allow closed form expressions for the weakly-nonlinear transfer of
electronic circuits [7] described by the Volterra kemels.
The Volterra kernels in the frequency domain: H,, are functions in s, the complex frequency. They can be seen as a generalization of the transfer function H ( s ) of a linear time-invariant
(LTI) system to a sum of transfer functions H , ( s i , . . . , sn) of a
time-invariant (TI) nonlinear system. In this representation, H I is
the traditional transfer function of the linearized system, HZ contains the nonlinearities ofthe second-order and so on.
The nth-order Harmonic Distortion at a certain frequency f
is defined as the ratio between the RMS output voltage at the nth
harmonic, obtained by imposing a tone of frequency f and unit
Figure 3: Simple small-signal model of the differential pair.
Figure 2: Typical configuration of a differential pair.
amplitude at the input of the circuit, and the R M S output voltage at
the base frequency (the desired transfer of the amplifier). Since the
input is in this case sinusoidal, the kernels simplify to a function
of one single frequency and the input amplitude A . The nth-order
harmonic distortion can be written as follows:
In (I) the higher-order terms due to mixing of the components
at positive and negative frequencies are neglected. Nevertheless
( I ) is a good approximation since for weakly nonlinear systems
the Volterra kemels decrease rapidly with increasing order, while
the higher-order terms are raised to the nih power. The Total Harmonic Distortion (THD) can be found by taking in the numerator
of (1) the sum of all Volterra kernels for n > 1.
3. T H E DIFFERENTIAL PAIR NONLINEAR MODEL
A drawback of the Volterra series is their tendency to complicate
for larger circuits, which will make the equations less transparent
and makes analytical treatment difficult. Therefore it is important
to take care in the choice of the model that will be used for establishing the kemels not to include unnecessary features. The models that will be presented in this section will be very simple, only
including the main nonlinearity and the most important dynamics.
In spite of these simplifications, the results given by this models
are surprisingly accurate (c.f. Sec. 5).
3.1. The basic model
A typical configuration of the differential pair, when it is used as
a non-inverting version of the CE stage is shown in Fig. 2. A very
simple nonlinear small-signal model for this circuit is shown in
Fig. 3. The signal source is modelled as a voltage source with a
series resistance R,. This resistance is in series with the base resistances T b of the transistors. In the following R will denote the sum
of these resistances. By assuming that R, varies between zero
and infinity, the whole spectrum from voltage-driven to currentdriven is covered. The transistors are assumed to operate in the
medium-current region and no (non-negligihle) mismatch is assumed, so all small-signal parameters are equal for both transistors. For the moment, the current source in the tail is assumed to
be ideal ( Y ( s )= 0).
The output of the model in Fig. 3 is given by:
where the hat denotes the AC part of a signal, the subscript t refers
to the tail current, p is the current-gain factor of the transistors
and VT = k T / q is the thermal voltage. The subscript bb is, here
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and in the following, short for blbz i.e. Ubb is the voltage over the
nonlinear input resistance as shown in Fig. 3.
Since the series expansion of a hyperbolic tangent has no evenorder terms, theoretically, the differential 'pair will not produce
even-order distortion. Therefore the second Volterra kernel does
not exist whereas the first and the third Volterra kemels are given
by:
For brevity the kemels are given already evaluated at a single input
frequency. The constants &denote the nih-order coefficientof the
expansion of (2) with respect to Ubb.
3.2. Modelling the second-order distortion
In practical implementations of differential pairs, a second-order
component is nearly always present. This can be due to mismatches between the transistors or other imperfections. A secondorder distortion of a completely different nature however can be
found in the case where the differential pair is used as a noninverting version of the CE stage, i.e. used asymmetrically.
If the tail-current source It in Fig. 2 has finite source admittance, the common mode voltage on the emitter of the transistors
will cause a current through this admittance. This effect was analyzed in [ 5 ] and in [8] a first-order approximation is given between
the common-mode input signal and this current. In both cases this
was done for MOS transistors and linked to the CMRR. The same
reasoning however can be applied here and taken in a more general
context. Adapted to BJTs the equation from [8] yields:
where ucmis the common-mode input signal and gm the transconductance of the transistors (hence the first-order approximation).
The admittance of the current source Y ( s )can he approximated by
a capacitor in parallel with a resistor, so Y (s) = sC, l / R o .The
tail current, which was assumed a constant bias current, tums out
to have a small variation superimposed. Since this current occurs
as a factor before the t a n h in (2), this relation should be rewritten
as:
+
--
D,f,ET."tid
Par
Toil-sUrrent
llOY7CD
where itis split into the constant current I , and the small varying
current i t
When the stage is driven asymmetrically (one side grounded),
and in this way used as a non-inverting version of the CE stage,
there is a direct relation between the (differential) input voltage
and the common-mode input voltage, ,U in (5). Consequently in
(6) both it and ubb are a function of the input voltage. Therefore
the last term in (6) will, instead of odd harmonics, only produce
even harmonics! Taking (6) as the goveming nonlinear function
completes the model of the differential pair and yields H z ( s ) :
When, on the other hand, a voltage source is taken, the source
resistance will be relatively small. In this case, the source resistance will short-circuit the input capacitance of the stage and the
distortion in the band depends only weakly on the frequency. Beyond the cut-off frequency, the distortion rolls of, as does the linear
in Fig. 4 plots
transfer of the stage. The solid line labelled HDJ,,
the distortion for a voltage-driven situation.
3.3. Distortion formulas
~.
. .. ............. . ...... (161
Combining (3), (4) and (7) with ( I ) gives the harmonic distortion
of the Differential Pair as a function of the frequency. The second
and third-order distortion thus obtained are more insightful when
rewritten in pole-zero-gain form:
IWI
-90
102
The pole p l and the zero z1 originate from the the input capacitance and small-signal resistance of the transistors. The pole p2
and the zero LZ originate from Y ( s ) .The gain constants, poles and
zeros are given by:
A
2K (1
Kz = -
1
+ 2gmRo)(1+ R / r x )
(10)
10'
10'
10'
10.
10'
Figure 4 Third-order distortion in a current and voltage-driven
differential pair. Simulations (dashed) and predictions by model
(solid). Maxima predicted by eqs. (16) and (17) indicated by dotted lines.
Considering the form of the distortion vs. frequency curves,
it is interesting for a designer to have an expression for the maximum of the curves. This allows one to know easily in an early
stage of the design which maximum distortion can be expected
and whether additional measures (e.g. power) are required. In the
current driven case this maximum is the value at the peak in the
characteristic:
ma~(HDa)l,,,~% 0.0558
where r,, = 2pV./It.is the small-signal linearized input resistance. As stated before, no mismatching between the transistors is
assumed so the small-signal parameters are equal for both transistors.
10'
(k)
(16)
The frequency at which this maximum is equals approximately
- 0 . 5 4 ~ ~In. the voltage driven situation, since the distortion does
not depend on the frequency, one can simply take the value at the
limit case for w = 0:
Both expressions depend solely on the value ;,/I,. This is the
output signal-to-bias ratio, i.e. the ratio of the AC output signal to
the DC bias current (both maxima are indicated in Fig. 4).
4. DISCUSSION AND SOME DESIGN RULES
4.1. The third-order distortion
The third-order distortion follows directly from the simple model
presented in section 3.1. It is very similar to the distortion found
in a CE stage [4]. It is possible to distinguish two qualitatively
different situations.
When the input is a current source, the source resistance R
will be relatively large. The distortion will strongly depend on
the frequency, since for lower frequencies the input capacitance is
an open circuit, whereas for increasing frequencies its impedance
decreases and its influence becomes noticeable. The solid line in
Fig. 4 plots the HDs,,,
where the subscript i denotes the currentdriven situation.
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4.2. The second-order distortion
The second-order distortion originates from the non-ideality of the
tail-current source. The non-ideality in the impedance of a good
current source is mainly capacitive. One would expect therefore
that the second-order distortion is mainly present for high frequencies, when the impedance of the capacitance will be low. For low
frequency the second-order distortion is indeed negligible as can
be seen from the solid curve in Fig. 5 . However towards the band
edge, this component becomes more important and (c.E Fig. 4
and 5 ) can even become larger than the third-order component [ 5 ] .
This can be of importance since it is near the band edge that the
242. The third-order distortion (HD3,,) is shown in Fig. 4. In this
situation the second-order distortion remains under -80dB.
6. CONCLUSIONS
Figure 5: Second-order distortion in a voltage-driven differential
pair, Simulations (dashed) and predictions by model (solid). Maximum predicted by eq. (18) indicated by dotted line.
loop gain of a negative-feedback amplifier decreases and ceases to
impose linearity.
For the current-driven example the second-order distortion remains extremely low. This is due to the fact that the location of
the pole p l is changed by a change in R, whereas the zero z1 is
unaffected.
For the voltage driven case, an expression can be derived for
the maximum value of the second-order distortion (peak of HD? .,
in Fig. 5):
Using a very simple model for a Differential pair, useful information can be obtained about the way distortion is generated in
this amplifier stage. This model, can also account for the secondorder component that occurs due to a non-ideal tail-current source
(CMRR). Simple expressions were derived for the second and thirdorder distortion for current and voltage-driving.
The well-known heuristic that increasing the bias current decreases distortion is confirmed. In particular the third-order distortion is quadratically inversely proportional to the bias current.
Another important datum is the way in which the stage is driven
or, equivalently, the relative value of the input-source resistance:
the higher the source resistance, the lower the Harmonic Distortion. This shows that there is a qualitative difference between the
distortion in a current-driven and a voltage-driven situation.
The differential pair exhibits theoretically no second-order distortion. However due to a non-ideal tail-current source it can have
second-order distortion when driven asymmetrically e.g. when it
is used as non-inverting version of the CE stage. In the CE stage
it is possible to cancel the third-order distortion due to the interaction of components originating from third- and second-order harmonics 141. In the differential pair this cancellation is not possible
since the second-order harmonics have a different origin and do
not interact with the third-order harmonics.
7. REFERENCES
The frequency at which the maximum is equals
i.e. the
geometric average of the two poles. The value of (1 8) also depends
on the output signal-to-bias ratio. On the other hand the quality of
the current source plays a role: if the capacitance COis small, the
distortion will be small.
5. VALIDATION
The predictions made are verified by comparing them to simulations with Pstar, a simulation program that uses the harmonic balance method.
For the simulations two identical QN3904 transistors were used
Icr, = 180,rb = 150 R). For the tail-current source R, = 10 MR
and CO= 5 pF were taken. The differential pair was simulated in
two different situations, a voltage-driven (R << rr) and a currentdriven ( R >> m)case.
For the voltage-driven case a voltage source with a source resistance of 100 Ci and an amplitude of 10 mV was used. Biasing
with200pAyields: C, = 13.7pF, C, = 2.4pF,gm = 3.9mS
and Rlr, = 0.0021. The effect of C, was modelled by taking it
in parallel to 6,.This is a reasonable simplification since simple
design resolutions exist for making this assumption true for situations where the Miller effect is non-negligible [I]. The results are
shown in dashed lines in Figs. 4 and 5; close agleement is obtained
between simulations and predictions.
For the current-driven case, the bias current was set at 500 pA.
The input source was a current source of 1 p A with a resistance of
R, = 5 MR. Small signal parameters of the transistors in this
case are: C, = 10.7pF, C, = 1.7pF, gm = 9.5mS. Rlr, =
1-264
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